GED Mathematical Reasoning: Introduction To Algebraic Expressions

  • Algebra is a branch of mathematics used to find unknown values. In algebra, we represent real life problems with mathematical equations and then solve them using a series of rules, processes and formulas.
  • One of the foundations of algebra is the idea of using a letter to represent a number. Specifically, a number we don’t know.  And we call this letter a “variable.”
  • Any letter can be used as a variable. Commonly used letters are:  x, y, n, m, a and b. And when a variable is used within the same problem, it represents the same value throughout.
  • An expression is a statement that contains numbers, variables, and operations like addition, subtraction, multiplication, and division. An expression may also contain parenthesis, exponents and square roots. Technically, an expression does NOT contain an equal sign.

For example: x - 4 is an algebraic expression.

  • Before you practice this concept on your own, it may good to review common key words and phrases used to imply certain operations. They are listed here under the operation they often (but do not always) imply.
Addition Subtraction Multiplication Division
plus less product divided by
combine subtract double divided into
more than difference triple quotient
add/added to less/more/farther than times goes into
increased by fewer of divided equally
sum decreased by twice per
total loss  multiply  each
gain minus    average
 altogether/in all take away    split
change

 

Example 1

Translate the words into an algebraic expression using “x” to represent the unknown value.

  1. Five times a number
  2. Three less than a number
  3. One-half the sum of a number and fiv

 

To translate “Five times a number” into an expression, I suggest taking it one word or phrase at a time.

The word FIVE translates into the number 5. The word “times” translates into multiplication. And the phrase “a number” can be represented by the variable “x” since that’s a value we don’t know. Putting this all together gives us the expression:   5 times “x”.

algeb22

Note how similar looking the multiplication sign and the variable “x” are.  We can remove the multiplication symbol and simply write this expression as: 5x. In general, when you see a number and a variable or more than one variable right up next to each other with no operation in between, the operation between them is assumed to be multiplication.

For the most part, translating a statement into a mathematical expression is pretty straight forward. There ARE one or two tricky phrases, though, that we need to be aware of.

Part (b) of the example illustrates one of these tricky phrases.

Given the statement:  Three less than a number. The word three translates into the number 3. The phrase “less than” implies subtraction. And the phrase “a number” can be represented using the variable “x” since that’s a value we don’t know.

It is tempting, then, to write the expression as:  3 - x

But if you think about it, the statement:  “three less than a number” implies that we START with a number and THEN subtract 3. Which means the statement “three less than a number” is actually translated as:  x - 3  So when you translate the phrase “less than,” it’s important to remember to flip the order from how the words appear in the statement.

algeb23

Now let’s consider part (c) of the example.

The phrase one-half translates into the fraction one-half. The word “of” implies multiplication. The word “sum” implies addition. The phrase “a number” translates into the variable “x” since it’s a number we don’t know. And the word five translates into the number 5.

algeb24

Now let’s talk about how this all fits together into an expression.

We are taking one-half of the sum of a number and five, which means we’re adding a number and 5 FIRST and THEN multiplying by one-half. To denote that the sum is found FIRST, we’ll use parenthesis around the sum since parenthesis is the first step in the order of operations. So we’ll write: one-half, open parenthesis, “x” plus 5, and then close the parenthesis.

one-half of the sum of a number and five:

\frac{1}{2}(x + 5)

 

Example 2

Peter, Elise, and Dominic equally split the profits of their business venture. If “x” represents the total profit earned, write the expression that represents each person’s share.

 

A first great step when solving an algebra problem is to identify the value that’s unknown and allow it to be represented by a variable. In this situation, the amount of total profit is unknown, so let’s let “x” equal the total amount of profit. It is a good to get in the habit of writing a “let” statement on your paper – stating what the variable represents. Our “let” statement for this example would be:

“Let ‘x’ equal total profit”

Now let’s talk about what operation is at work here.

Since we are SPLITTING an amount, this implies the operation of division. Additionally, we are splitting the total profit 3 ways so we will be dividing the profit by 3. Putting this all together, the total profit divided by 3 represents the amount that each person will receive. This idea can be represented by the expression: “x” divided by 3.

x \div 3

It is typically desirable in algebra to represent division using a fraction so we will represent “x” divided by 3 as “x” over 3.

\frac{x}{3}

Before we move on to our last example, I want to share with you a little something that will be helpful as you move forward in your learning of algebra.

The expression “x over 3” can be rewritten as a product. In other words:  “x” over 3 can be rewritten as:  one-third “x”. Here’s why: If we work backward, one-third “x” can be re-written with the multiplication sign as:  one-third times “x”. Next, we can write “x” as a fraction by putting it over 1.

\frac{1}{3} \times x = \frac{1}{3} \times \frac{x}{1}

And from there, to multiply factions we know to multiply straight across. One times “x” is just “x” (since anything times one is itself) and 3 times 1 is 3. Making the end result:  “x” over 3

\frac{1}{3} \times \frac{x}{1} = \frac{1 \times x}{3 \times 1} = \frac{x}{3}

 

Example 3

The cost to rent a vehicle at A+ Car Rental is $100 plus $35 per day. If x represents the number of days a vehicle is rented, write an expression that could be used to find the total cost of renting a vehicle from this company.

 

In example three, the total cost equals $100 plus $35 for each day. So if the car is rented for one day, the total cost is: 100 plus 35 times 1. If the car is rented for two days, the total cost is 100 plus 35 times 2. If the car is rented for three days, the total cost is 100 plus 35 times 3. If the car is rented for “x” days, the total cost is 100 plus 35 times “x.”

The expression that represents the total cost of renting a car from the A+ Car Rental company is: 100 + 35 \cdot x

As you can see from the strategy used to determine the expression for example 3, sometimes it is helpful in algebra to let the unknown value equal a specific number. As we change the number we can look for a pattern, which ultimately leads to our answer.

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